\(\mathbb{R}^n\) with \(d(\vec{x}, \vec{y}) = \sqrt{(x_1 - y_1)^2 + \dots + (x_n - y_n)^2}\), also known as the Euclidean metric on \(\mathbb{R}^n\), which is the implicit metric on \(\mathbb{R}^n\) if left unspecified.

For a space of continuous bounded functions on \(\mathbb{R}\), denoted \(\mathcal{C}_b(\mathbb{R})\), \(d_{\sup}(f, g) = \sup_{x \in R} |f(x) - g(x)|\), also known as sup norm. The supremum exists because the functions are bounded in their values.